Probability distributions, especially in applications, are generally specified via density functions; alternative representations, including the characteristic function, moment generating function, and sequence of moments, are most commonly encountered in theoretical settings. These alternative means of specification do, however, give rise to the construction of certain approximations that can facilitate the implementation of likelihood methods or the calculation of probabilities, even when the density is not available in closed form. The feasibility of a unified treatment of this topic stems from a number of properties shared by probability transforms in general.
"Approximate Likelihood and Probability Calculations Based on Transforms." Ann. Statist. 12 (3) 989 - 1005, September, 1984. https://doi.org/10.1214/aos/1176346716